A Bibliometric Visualized Analysis and Classification of Vehicle Routing Problem Research

Abstract

The vehicle routing problem (VRP), as a classic combinatorial optimization problem, has always been a hot research topic in operations research. In order to gain a deeper understanding of the VRP problem, this work uses the knowledge graph to comprehensively analyze and summarize the literature related to VRP from 1959 to 2022 in the Web of Science (WoS) database. Firstly, according to the basic statistical information of the literature, the annual publications, the authors, their institutions and countries, the keyword co-occurrence, and the literature co-citation network are analyzed to comprehensively summarize and generalize the research on VRP and predict its future development trend. The results show that, in the past 60 years, there have been abundant changes in the research on VRP. The United States and China have made the most important contributions in the field of VRP. According to the WoS literature retrieval results and classification methods, the VRP models and their solutions are comprehensively classified, and the model solving algorithms are divided into exact algorithms, heuristic algorithms, metaheuristic algorithms, hyper-heuristic algorithms, machine learning, etc. The results show that the development of information computing technology plays an important role in research on the VRP problem, and dynamic VRP, hyper-heuristic algorithms, deep reinforcement learning, etc. are the future development directions of the VRP model and its optimization. The results of this research can provide help and guidance for beginners and scholars outside the industry to comprehensively understand the development and research hotspots of VRP.

1. Introduction

The vehicle routing problem (VRP) first evolved from the traveling salesman problem (TSP), which is a classic combinatorial optimization problem in operations research. Since its inception in 1959 [1], it has been one of the most fundamental problems in network optimization, attracting extensive attention from scholars due to its wide applicability and economic significance, and it has quickly been applied to fields such as operations research, management science, and computer science.

Unlike the traveling salesman problem, VRP considers constraints such as delivery time windows and route lengths, designing the best route for delivery vehicles. The classic VRP can be described as follows: under certain constraints, how does the supplier arrange vehicles to deliver goods from one or more depots to customers with known geographical locations, so as to minimize the total delivery cost. VRP can directly reduce the number of vehicles used under certain constraints, effectively reduce the transportation cost, improve the quality of customer service and the core competitiveness of the enterprise, and reduce the emission of CO₂ and protect the environment. However, the reality is very complicated, and more and more constraints are introduced into the basic VRP, such as vehicle capacity, number of depots, delivery time windows, allowable trips, and demand nature. The research objectives and objects have changed, and the problem has evolved from the original simple vehicle arrangement scheduling problem to a complex system problem.

Since Dantzig and Ramser first introduced the VRP and its solution in 1959, the VRP model and its solving algorithm have been extensively researched [1]. After more than 60 years of development, with the deepening of the research on the path problem and the complexity of the real path problem, many different extended and changed forms have been produced in academic research on and the practical application of the vehicle route problem, with more and more constraints being added to the basic vehicle route problem, thus making different types of variant vehicle route problems more realistic; examples include the uncertain demand VRP [2], the dynamic VRP [3,4], and the green VRP [5,6,7]. Due to the complexity of the constraints in the real-world VRP problem, many VRP optimization-solving algorithms have emerged for different VRP models, including exact methods, heuristic, metaheuristic methods, and hyper-heuristic algorithms. For example, Barma et al. proposed a bi-objective CVRP based on priority for two types of customers and developed a hybrid metaheuristic algorithm based on the greedy randomized adaptive search procedure and nondominated sorting genetic algorithm to solve the model; they then analyzed the results according to three scenarios of average delay calculation based on customer type. The results showed that the hybrid GRASP-NSGAII algorithm outperformed the NSGAII algorithm, and the proposed model could be extended to more than two types of customers with different priorities [8]; A hybrid multi-objective evolutionary algorithm through a cluster primary route/secondary approach was proposed by Dutta et al. to solve the multi-objective open green vehicle routing problem in sustainable environments, which uses the extended strength Pareto evolutionary algorithm (SPEA2) and nondominated sorting-based genetic algorithm (NSGA-II) to search for the best sub-route, and then uses the VIKOR method to identify the decision maker’s choice-based solution for each cluster [9]. The effectiveness and feasibility of the proposed model were demonstrated through numerical examples and statistical analysis, with SPEA2 showing better performance than NSGA-II; this study provided a novel and effective solution for the open green vehicle routing problem. In addition, with the breakthrough progress of machine learning in multiple fields such as computer vision and natural language, it has also received widespread attention in VRP solving. For example, using deep reinforcement learning technology to improve metaheuristic algorithms and utilizing deep reinforcement learning technology in hyper-heuristic algorithms could effectively improve the quality, convergence speed, and robustness of solutions [10,11,12,13,14,15].

VRP and its variants were originally reviewed by Golden on the basis of several high-quality papers [16]; subsequently, different review articles have explored different variants of VRP and the evolution of its solving algorithms [17,18,19,20,21]. Tan et al. used a classification framework to summarize VRP and its solving algorithms [19]. With regard to the classic VRP, Zhang et al. classified it according to its characteristics and practical applications, and then gave a unified description and mathematical models of each type of problem [21]. However, for the classification of VRP model solving algorithms, the existing research only includes exact algorithms, heuristic algorithms, and metaheuristic algorithms, whereas it does not summarize and classify the super-heuristic algorithms and machine learning algorithms developed in recent years. With the increasing application of hyper-heuristic algorithms and machine learning algorithms in combinatorial optimization, especially the advantages of deep reinforcement learning technology in the solution of VRP problems, it is necessary to classify and summarize the application of hyper-heuristic algorithms and machine learning algorithms in VRP solutions [13,14,15].

With the in-depth development of VRP research, the number of publications on VRP has increased rapidly, making it difficult to track such massive knowledge. Bibliometric methods, based on mathematical and statistical methods, can effectively quantify the analysis of massive scientific literature data, analyze the evolution of scientific problems over time, and determine the knowledge acquired and its boundaries [22]. Using visualization methods, knowledge maps of bibliometric statistics results can be obtained, which can intuitively display information such as the evolution, hotspots, and trends of related research fields, helping us to gain a comprehensive understanding of the past, present, and future trends of research topics [23].

It is very important to review and summarize existing knowledge to guide and help find new research directions in VRP. In order to obtain a systematic and detailed understanding of the current situation and future development direction of VRP from the literature, we used a literature metric knowledge map method to conduct a comprehensive system analysis of VRP based on the literature in the WoS database from 1959 to 2022. Unlike previous algorithm taxonomy studies, we added hyper-heuristic algorithms and machine learning algorithms that were not previously considered, and then summarized and classified them according to the current research status and characteristics, so as to obtain comprehensive knowledge and understanding of the VRP solving algorithm. The purpose of this study was (1) to obtain the attribute changes of the published literature, such as publication time and region, (2) to provide an accurate description of the evolution of VRP research over time, (3) to visualize the literature using knowledge maps and analyze the statistical results using network analysis to obtain the intrinsic relationship between the literature and authors, (4) to obtain a detailed classification and understanding of the current VRP model and its solution algorithm, and (5) to conduct a comprehensive tracking, analysis, and discussion of the current status, frontiers, and future trends of VRP research. We hope that this article can provide a comprehensive and appropriate overview of VRP research in the past 60 years, helping new researchers to gain an understanding of this field, as well as promoting the development of VRP research.

2. Data and Methods

2.1. Data Acquisition

The WoS database is composed of SCIE, SSCI, CPCI, and the Arts and Humanities Citation Index citation index, which includes more than 9000 scientific journals; it can retrieve relevant information of the articles included and cited, search for early and recent academic information related to the topic, and provide detailed abstracts, citations, etc. The WoS core literature includes 10 sub, eight citation, and two chemical databases, while the core citation library includes SCI, SSCI, and A&HCI; the data are updated weekly. The core citation library constitutes a set of academic journals that have been strictly evaluated and screened worldwide, making WoS both a literature retrieval tool and an important basic evaluation tool for literature metrics and science metrics.

In this study, the WoS database was used to retrieve VRP literature, using the WoS core collection; the topic was set as “vehicle routing problem”, the document type was set as “article” or “review”, the language was set as “English”, and the publication time was set as “1959–2022”. After that, the whole retrieval record was checked, and some obviously irrelevant data were removed; finally, 10,321 valid records were obtained. The final retrieval record may have still contained some invalid data, but these were revealed and corrected in the subsequent knowledge graph, while also providing additional meaningful discoveries [23]. WoS retrieval records provide relevant information such as the abstract, keywords, authors, countries, institutions, titles, the number of citations of the article, and its citation information (

Table 1. Basic information of the WoS dataset in the field of VRP from 1959 to 2022.

Type	Number
Publications	10,321
Categories	144
Authors	17,618
Countries	106
Affiliations	4472
Titles	1000
Citing articles	95,622
Time Cited	205,779
H-index	202

). The abstract and keywords of the article can help us quickly understand the theme and content of the article. The number of times an article has been cited, as well as other articles that cite it, can help us understand the impact and importance of that article in academia. Information such as the author, their institution, and the research field reflect the author’s background and professional field. Using this information, we can obtain the topic, author, region, publication journal’s influence, and academic level of the article. In addition, the research hotspots and trends in this field, as well as the connection and influence degree among related research results, can be obtained through the analysis of keywords and citations.

2.2. Scientometric Analytical Methods

Bibliometric analysis utilizes mathematical and statistical methods to quantitatively analyze the relationship among books, studies, and other information resources, evaluate the current research status and trends of certain research fields, and find new research directions, being widely applied in various fields [22]. Statistical analysis of information such as authors, institutions, and keywords can provide a quick understanding of the development of the literature and its trends [23]. Through statistical, clustering, and network analyses of literature attribute characteristics, software such as CiteSpace and VOSviewer can provide a visual knowledge map of multidisciplinary literature research, track the connections among different clusters in related topic research fields, and identify research progress, new trends, and new dynamics in specific research fields in scientific development [24,25]. In this paper, CiteSpace was used to visualize the VRP literature in the WoS database and summarize its research status, hotspots, and evolution.

3. Results

3.1. Basic Characteristics of the Bibliometric

3.1.1. Temporal Evolution of Publications and Citations

The time series distribution of research papers and their citations can reflect the re-search status and trend of a certain research topic in this period [26]. Figure 1 shows the number of papers and their citations related to VRP retrieved from the WoS database from 1990 to 2022. In the past 30 years, the number of papers published and their citations increased significantly, indicating that VRP research is getting more and more attention. From the figure, it can be seen that, before 2008, the number of papers published each year was less than 200, and the citation rate was relatively low, indicating that the research attention at this stage was relatively limited and in a slow growth stage; from 2009 to 2017, the number of papers published gradually increased but remained below 600, whereas the citation rate exceeded 5000 in 2009, and reached 22,336 in 2018, showing that VRP-related research gradually received increasing attention; since 2018, the number of papers published each year has exceeded 700 and has grown rapidly, while the citation rate has also surged to 446,346. The number of papers published and the number of citations are expected to continue increasing, indicating that VRP research is attracting more and more attention from scholars, which is also related to the rapid development of the logistics industry and related fields in recent years, as well as the urgent needs of the carbon peak [5].

3.1.2. Research Area and Journal Source

According to the basic information retrieved from the WoS Core Collection, it was found that nearly 144 subject categories of literature published in the WoS were related to VRP research, indicating that VRP research has a wide range of research fields. The five categories with the highest number of papers were operations research management science (ORMS), transportation science technology (TST), electrical and electronics engineering (EEE), computer science interdisciplinary applications, and industrial engineering (Figure 2). These categories had more than 1000 papers related to VRP research. Operations research management science was the largest category, with more than 4353 papers, accounting for about 42.5% of the total. The United States and China were the main countries contributing to the three most popular subject categories, while China published the most VRP papers related to TST, which is closely related to the rapid development of China’s economy and computer information technology (

Table 2. Top three countries in terms of publications in the top three categories.

Categories	USA	China	Canada
ORMS	893	629	628
TST	650	563	209
EEE	264	623	73

).

According to the analysis results of WoS Core Collection retrieval, there were 901 publications related to VRP research, including journals, conference proceedings, and books. Figure 3 shows the top 20 publication sources ranked by the number of publications. The European Journal of Operational Research (EJOR) and Computers Operations Research (COR) were ranked highest, with more than 500 papers published. Transportation Science (TS) and Computers Industrial Engineering (CIE) followed closely, with 339 and 336 papers published, respectively. Transportation Research Part E: Logistics and Transportation Review (TRE) ranked fifth. Canada published the most papers in EJOR and COR. The papers published by the United States in TS accounted for about one-third of the total, while Chinese papers related to VRP were commonly published in TRE, accounting for about 36% of the total (

Table 3. Top five countries contributing to the top five publication sources.

Publication Titles	Counts	% of Countries Contribution
EJOR	629	Canada—16.06	Germany—14.94	USA—13.99	France—11.76	China—10.02
COR	575	Canada—20.87	USA—12.86	Italy—11.48	France—10.96	Brazil—10.09
TS	339	USA—34.81	Canada—23.01	Italy—16.81	Germany—15.04	Netherlands—10.32
CIE	336	China—23.51	USA—17.56	Iran—10.71	S Korea—8.33	Turkey—8.04
TRE	219	China—35.55	USA—24.61	Canada—10.16	France—6.64	England—5.86

). Therefore, the research results of VRP are usually published in journals related to management and computers, such as EJOR and COR, while some are included in relevant conference proceedings, such as the IEEE International Conference on Automation Science and Engineering (IEEE CASE).

3.1.3. Geographic Distribution of Publications

As shown in Figure 4, a total of 108 countries and regions have conducted research on VRP, with China (3003, 29.33%), the United States (1942, 18.97%), Canada (885, 8.64%), Italy (586, 5.72%), and France (577, 5.63%) representing the top five countries in terms of the number of publications. European countries such as the United Kingdom, Germany, and Spain were highly ranked in terms of the number of publications, followed by Asian countries such as Iran, India, South Korea, and Singapore. Chinese research papers on VRP accounted for one-third of all papers, and the number of papers increased rapidly from 10 in 2008 to 559 in 2022, while the number of citations also increased rapidly from 92 in 2008 to 12,281 in 2022, reflecting the increasing attention of Chinese scholars to VRP, leading to more results.

3.2. Collaboration Network Analysis of Bibliometric

3.2.1. Author Collaboration Network

According to the WoS literature retrieval record, the number of papers, the number of citations, and the h-index of authors in VRP research could be obtained. The h-index mainly represents the scientific productivity and influence of scientific researchers [27]. During the period from 1959 to 2022, there were 17,618 authors in the retrieval dataset of VRP research, 10 of whom published more than 50 papers.

Table 4. Top 10 published authors, number of publications, mean citations per article, and h-index.

Rank	Authors	Publications	Times Cited		H-Index
			Total	Average per Term
1	Laporte, G	202	16,979	87.50	72
2	Gendreau, M	115	10,536	94.24	51
3	Wang, Y	78	1121	16.51	20
4	Juan, AA	74	1576	27.61	29
5	Cordeau, JF	65	7930	124.15	39
6	Speranza, MG	61	2918	49.82	32
7	Archetti, C	58	2333	42.19	28
8	Hartl, RF	56	3302	60.52	29
9	TarantiliS, CD	54	2179	43.13	32
10	Tavakkoli, R	53	1357	26.68	20

gives the top 10 authors, along with their citation number and h-index. Among them, Laporte from the United States had nearly twice the number of articles as the second author, with his h-index also being ranked first; Gendreau occupied second place in terms of the number of articles and h-index; Cordeau, who ranked fifth in terms of the number of articles, had a total of 7930 citations (excluding self-citations), and his h-index was also 39; Wang from China ranked third in terms of the number of articles, but only had 1121 total citations, which is far lower than the citation number of the top ten authors, thus reducing his h-index. The remaining authors had an h-index between 28 and 32, and their productivity and influence were relatively balanced.

The author collaboration co-occurrence network reflects the cooperation relationships of scholars in a study field [25]. Following CiteSpace visual analysis, a cooperation network with 1086 nodes and 759 links was formed (Figure 5). The nodes represent the objects of analysis (authors and institutions in the literature), the size of the node circle represents the number of authors’ articles, the larger the size of the circle represents the number articles (larger nodes indicate a higher frequency of occurrence), and the color and thickness of the node circles represent the frequency of the articles in different time periods. The lines between nodes represent the cooperation relationship and strength of the analysis objects [25]. As shown in Figure 5, different research groups were formed around the most productive and influential authors, promoting exchanges and cooperation among different research groups. For example, the research groups of Juan and Hartl were connected through different modes centered around Laporte, Cordeau, and Gendreau, while further communicating and cooperating with other research teams, which effectively promoted knowledge flow in VRP research.

3.2.2. Country Collaboration Network

The country network analysis reflects the cooperative relationships among countries in a research field [25]. According to the CiteSpace statistics, a co-occurrence network composed of 106 nodes and 56 edges was formed. It can be noted that the countries and regions with higher productivity had close collaboration (Figure 6). In terms of VRP research, it is worth mentioning that the UK (0.88), Scotland (0.78), and Finland (0.75) had high centrality and intermediacy, playing an essential role in connecting with other countries.

3.2.3. Institutional Collaboration Network

Figure 7 shows the collaboration network map of VRP research from 1959 to 2022. According to the WoS retrieval subset, a total of 4472 institutions contributed to VRP research. The Universities of Montreal, HEC Montreal, and Polytechnique Montreal in Canada were high-producing institutions for VRP research, publishing 559, 266, and 189 articles, respectively. The National Center for Scientific Research in France and Beijing Jiaotong University in China ranked fourth and fifth, publishing 179 and 145 articles, respectively, followed by Hong Kong Polytechnic University (131), National University of Singapore (124), and University of Bologna in Italy (120). These institutions conducted extensive exchanges and cooperation in VRP research, effectively promoting its further development.

3.3. Research Base, Hotspots, and Frontiers

3.3.1. Document Co-Citation Network Analysis

When two papers are cited by one or more subsequent papers at the same time, they form a co-citation relationship, which can form a cluster network space of a specific research topic; the cited papers constitute the research foundation, while the corresponding citations represent the research frontier [28]. The co-citation relationship among papers changes over time; through an analysis of the co-citation network of papers, the knowledge structure, evolutionary dynamics, and paradigms of a research field can be explored. If the co-citation frequency of two papers is higher, it indicates that the research contents of these two papers are more similar. In a co-citation network, the betweenness centrality measure can be used to highlight the key points representing the potential paradigm changes [23,25,29]. CiteSpace adopts betweenness centrality to discover and measure the importance of papers [25].

Figure 8 shows the co-citation network map of the related research papers on VRP from 1959 to 2022. According to the CiteSpace visualization analysis results, the co-citation network of papers consisted of 2383 nodes and 3767 links, the modularity value of the network was 0.8896, the mean silhouette value was 0.9502, and the harmonic mean (Q, S) was 0.9189 [25]. These results show that the co-citation network obtained was of high reliability. As can be seen from the figure, Braekers (2014), Toth (2014), Lin (2014), Schneider (2014), and Hiermann (2016) were the top five papers, with co-citation frequencies of 140–202. The high frequency of citations in the network indicates the importance of these papers in VRP research.

Simple co-citation analysis can only observe the relationship among documents, but cannot discover the relationship among topics discussed in text. Co-citation clustering can organize the keywords in documents and analyze the themes represented by each cluster from the co-citation network. The sum of the core document set of the co-citation clustering and the latest documents citing these core documents represent the research frontier [28,30]. Figure 9 shows the clustering result obtained from the co-citation network, producing 26 clusters (

Table 5. The largest 26 clusters of co-citation network.

ID	Size	S	Label (LLR)	Top Items	Average Year
0	147	0.871	Open vehicle routing problem	Open vehicle; capacitated vehicle; mix vehicle; routing problem; electric vehicle	2006
1	137	0.923	Stochastic vehicle routing	Stochastic demand; robust vehicle; routing problem; local search; electric vehicle	2011
2	132	0.959	Parallel algorithm	Parallel evolutionary algorithm; tabu search heuristic; parallel tabu search; bi-objective vehicle; planning model	1995
3	128	0.919	Metaheuristic	Iterated local search algorithm; efficient metaheuristic algorithm; collecting vehicle; routing product; dehydration plant	2001
4	118	0.984	Electric vehicles	Electric vehicle; partial recharge; electric bus; mixed charging rate; ambient temperature	2017
5	98	0.953	Two-echelon VRP	Capacitated location-routing problem; two-echelon capacitated vehicle; two-echelon vehicle; consistent vehicle; location-routing problem	2012
6	97	0.966	Fuel consumption	Green vehicle; cold chain logistics; urban area; fuel consumption; pollution routing problem	2015
7	92	0.921	Home healthcare	Home health care; resource sharing; routing problem; strategic oscillation; transportation resource sharing	2015
8	85	0.998	Drones	Multiple drone; traveling salesman problem; routing problem; last-mile delivery; parcel delivery	2018
9	80	0.976	Arc routing	Capacitated arc; split delivery vehicle; lean production system; annotated bibliography; recent result	2005
10	73	0.926	Column generation	Dial-a-ride problem; dynamic transportation; pooling problem; introducing heterogeneous user; adaptive insertion algorithm	2006
11	53	0.973	General interarrival times	Euclidean plane; dynamic vehicle-routing problem; new generation; robust algorithm; addressing uncertainty	1991
12	50	0.958	Dynamic vehicle routing	Dynamic pickup; dynamic vehicle; hybrid adaptive predictive control; delivery problem; waiting strategies	2002
13	46	0.960	Inventory routing	Inventory-routing problem; benders decomposition; vendor-managed inventory; reverse logistics; stochastic inventory-routing	2012
14	45	0.988	Two-echelon production routing	Asymmetrical capacitated vehicle-routing problem; distance-constrained vehicle-routing problem; capacitated vehicle-routing problem; exact algorithm; combined vehicle-routing	1981
15	44	0.950	Three-dimensional loading	Multiple stack; loading constraint; traveling salesman problem; double traveling salesman problem	2010
16	36	0.964	Multi-trip	Release date; same-day delivery; multiple trip; community logistics; stochastic dynamic vehicle	2015
17	33	0.980	Regression analysis	Stochastic customer demand; probabilistic analyses; central inventories; vehicle-routing cost; two-echelon distribution system	1994
18	27	1.000	Bike sharing	Rebalancing problem; bike-sharing system; bike sharing; repositioning problem; multiple vehicle	2015
19	25	0.979	Team orienteering problem	Orienteering problem; solving tourist trip design problem; capacitated team; algorithmic approaches; LP-based granular variable neighborhood search	2010
20	22	0.983	Occasional drivers	Occasional driver; delivery option; last-mile vehicle; collaborative urban transportation; crowd-shipping setting	2017
22	17	1.000	Route construction and improvement	Vehicle-routing; algorithm; scheduling problem; routing problem; electric vehicle	1982
26	14	0.998	Lagrangian relaxation for delivery problems	Capacitated vehicle-routing; unique item; multi-stop multiterminal delivery route; parallel saving; vehicle-routing problem	1988
31	8	0.987	Perishable products	Perishable product; robust bi-objective model; sustainable inventory; two-echelon inventory; cold-chain inventory	2016
39	6	0.992	Constraint programming environment	Constraint programming environment; global search; learning hybrid algorithm; single vehicle pickup; tabu search heuristic	1997
43	5	1.000	Branch and cut	Cut; parallel branch; capacitated vehicle; routing problem; electric vehicle	1998

). Cluster #0 was the largest cluster, containing 147 nodes, with a single coefficient value of 0.871; the keywords labeled by LLR (log-likelihood ratio) mainly included “open vehicle; capacitated vehicle; mix vehicle; routing problem; electric vehicle, etc.”, and the main research topic was the open vehicle routing problem model and its variants. Cluster #1’s research was mainly related to “stochastic demand; robust vehicle; routing problem; local search; electric vehicle, etc.”. Cluster #2 mainly included “parallel evolutionary algorithm; tabu search heuristic; parallel tabu search; bi-objective vehicle; planning model”. Cluster #3 involved the application of metaheuristic algorithms for VRP models, featuring “iterated local search algorithm; efficient metaheuristic algorithm; collecting vehicle; routing product; dehydration plant, etc.” Cluster #4 was related to VRP problems in electric vehicles, featuring “electric vehicle; partial recharge; electric bus; mixed charging rate; ambient temperature, etc.”. Other clusters included TSP, dial-a-ride problem, inventory-routing problem, location-routing problem, and variants of VRP models and their algorithms: “dynamic vehicle-routing problem, asymmetrical vehicle-routing problem, capacitated vehicle-routing problem, combined vehicle-routing, multiple drones, green vehicle VRP, stochastic inventory-routing, etc.”. According to the clustering results, the variants of open VRP, stochastic VRP, dynamic VRP, two-echelon VRP, electric VRP, drones VRP, multi-trip VRP, and Green VRP are the hotspots of current VRP variant research, while metaheuristic algorithms commonly used for optimizing VRP models.

According to the co-cited clustering results (

Table 6. Top 10 co-citation literatures.

Cluster ID	Citation Counts	Reference
1	203	Braekers K, 2016, COMPUT IND ENG, V99, P300 [31]
15	163	Toth P, 2014, MOS-SIAM SER OPTIMIZ, V0, P1 [32]
8	158	Dorling K, 2017, IEEE T SYST MAN CY-S, V47, P70 [34]
4	153	Montoya A, 2017, TRANSPORT RES B-METH, V103, P87 [36]
6	150	Lin CH, 2014, EXPERT SYST APPL, V41, P1118 [40]
4	146	Hiermann G, 2016, EUR J OPER RES, V252, P995 [37]
4	140	Schneider M, 2014, TRANSPORT SCI, V48, P500 [38]
8	139	Agatz N, 2018, TRANSPORT SCI, V52, P965 [33]
8	125	Wang XY, 2017, OPTIM LETT, V11, P679 [35]
4	124	Desaulniers G, 2016, OPER RES, V64, P1388 [39]

), the most cited work was Braekers (2016) (203 times) in Cluster #1. Braekers et al. classified 277 VRP papers published from 2009 to 2015 using the comprehensive classification method of Eksioglu et al., and then analyzed the trend of VRP research [31]. The second most cited work was Toth P (2014) (163 times) in Cluster #15. Toth et al. introduced the vehicle routing problem (VRP) and its various variants, ranging from the basic capacitated vehicle routing problem to more complex vehicle routing problems with time windows, pickup-and-delivery problems, stochastic vehicle routing problems, dynamic vehicle routing problems, etc. [32]. There were three highly cited papers in Cluster #8, Dorling (2017), Agatz (2018), and Wang (2017), which mainly studied the UAV truck problem related to truck scheduling, enabling effective cooperation between trucks and UAVs in the last-mile delivery and emergency response scenarios, while taking into account the impact of UAV battery and load weight on energy consumption; heuristic algorithms were used to effectively improve the UAV vehicle solution to save delivery cost and time [33,34,35]. There were four highly cited papers in Cluster #4, namely, Montoya (2017), Hiermann (2016), Schneider (2014), and Desaulniers (2016). These papers studied the electric fleet scale and mixed-vehicle delivery problem (EFSMFTW) with time windows and charging stations. Montoya (2017) et al. expanded the current E-VRP model to consider the nonlinear charging function, and then proposed a hybrid metaheuristic method. The results showed that ignoring the nonlinear charging may lead to infeasible or too expensive solutions [36]. Hiermann (2016) et al. proposed a hybrid heuristic method by considering the constraints of customer location time windows. By using a precise method to solve a set of newly created EFSMFTW and existing single vehicle type benchmark test instances, the effectiveness of the proposed method was demonstrated [37]. Schneider (2014) et al. introduced the electric vehicle routing problem with time windows and charging stations (EVRPTW), and then combined the variable neighborhood search algorithm with the tabu search heuristic, constrained by vehicle freight capacity and customer time window. A hybrid heuristic method was proposed, and the test on the newly designed EVRPTW instances and related benchmark instances showed the high performance and positive effect of hybridization of the proposed heuristic method [38]. Desaulniers (2016) et al. proposed an exact branch, price, and cut algorithm for four variants of the electric vehicle routing problem with time windows, and then generated feasible vehicle routes through customized monodirectional and bidirectional labeling algorithms. It was found that, compared with the variants of single and full charging, allowing multiple and partial charging could help reduce the route cost and the number of vehicles used [39]. Such publications, along with other highly cited papers in the clustering network, constitute the current research hotspots of VRP.

3.3.2. Keywords Co-Occurrence Network Analysis

Keywords are highly summarized topics of literature, and statistical analysis of the frequency and centrality of keywords can reveal the research hotspots and development trends in a field. Co-occurrence analysis of keywords utilizes the co-occurrence of keywords to determine the relationship among various topics in the field [41,42,43]. The node selection method selected was the g-index algorithm; after trimming, a keyword co-occurrence network map with 324 network nodes and a network density of 0.0072 (Figure 10) was obtained. The number of nodes denotes the number of trimmed network keywords, the size of the circle represents the frequency of the keywords (a larger circle indicates a higher frequency), and the thickness of the line represents the degree of closeness [26,44].

As shown in Figure 10, the node of keyword “vehicle routing problem” was the largest, indicating that this word appeared most frequently in the literature, consistent with the attention of the research field. The second most frequent keywords were algorithm, optimization, model, time window, tabu search algorithm, genetic algorithm, system, traveling salesman problem, heuristic algorithm, etc., representing the hotspots of VRP research. The intermediate centrality represents the position of a keyword among all keywords [25]. Generally speaking, a higher frequency indicates a higher intermediatory centrality of a keyword; examples include algorithm (2770, 0.47), vehicle routing problem (3874, 0.43), optimization (1558, 0.35). However, frequency and centrality do not exhibit one-to-one correspondence, whereby the centrality of keywords with lower frequencies may be larger, while the centrality of keywords with higher frequencies may be smaller examples include cost (120, 0.84), tabu search algorithm (903, 0.1), genetic algorithm (870, 0.06), green logistics (44, 0.33), search (598, 0.02), and fleet size (62, 0.34). Nodes with intermediate centrality greater than 0.1 play the role of bridges in the network, indicating the transformation of research hotspots, as well as their dynamics in the research field [25]; for example, there was a transformation from the traveling salesman problem node to the VRP node, and then to the algorithm node, and finally to the heuristic algorithm node, reflecting the evolution of VRP research from TSP to further in-depth research, including the evolution of models and optimization algorithms.

Keyword clustering analysis aggregates closely related keywords together to form clusters, which helps to establish the knowledge structure and gain an overall understanding of research in a field [45,46]. On the basis of the keyword co-occurrence graph, through repeated adjustment of the threshold, the high-frequency keywords of the research literature were clustered using the log-likelihood algorithm; the network clustering module value Q = 0.8534 (>0.3) and clustering weighted average silhouette value S = 0.9528 (>0.7 and close to 1) indicate that the clustering network reliability was high and the similarity was sufficient. A total of 17 clusters (Figure 11) were obtained. A smaller cluster number indicates that more keywords were contained in the cluster. Cluster #0 contained keywords ”transportation; logistics; allocation; dynamic programming; impact; cost; operation; city logistics; stochastic demand; policy; split delivery; stochastic programming; decomposition; urban freight transport; allocation; horizontal cooperation; cost; operation, etc.”; Cluster #1 mainly focused on optimization methods, including keywords ”optimization; genetic algorithm; ant colony algorithm; particle swarm optimization; combinatorial optimization; simulated annealing; evolutionary algorithm; memetic algorithm; capacitated vehicle routing problem, etc.”; Cluster #2 mainly reflected the intelligent transportation network problem, including keywords ”vehicular ad hoc networks; routing protocols; mathematical model; protocol; VANET; machine learning; neural network; intelligent transportation system”; Cluster #3 mainly comprised research on the traveling salesman problem, with keywords ”vehicle routing; vehicle routing problem; approximation algorithm; path planning; last-mile delivery; drone delivery; urban logistics; vehicle scheduling, etc.”; Cluster #4’s keywords were mainly related to pickup and delivery related research, including ”dynamic vehicle routing; dial-a-ride problem; pickup and delivery problem; large neighborhood search; neighborhood search; robust optimization, etc.” From the above analysis, it can be seen that the research hotspots obtained from keyword co-occurrence clustering analysis were basically consistent with the research hotspots reflected by the highly cited literature.

A sudden increase in frequency of a keyword in certain years can effectively reflect the hotspot changes and evolutionary trends of a research field [44,47]. According to the keyword clustering map, the top 25 keywords were obtained using keyword burst detection (Figure 12). As can be seen from the figure, the keyword with the strongest intensity was the “tabu search algorithm”, followed by “vehicle routing problem, traveling salesman problem, algorithm, column generation, scheduling problem, etc.”. According to the time node division, the burst words from 1976 to 2020 were “traveling salesman problem, heuristic algorithm, scheduling problem, algorithm, time window, combinatorial optimization, tabu search algorithm, integer pro-gramming, and column generation”, among which ”heuristic algorithm” had the longest duration, representing a main research topic with great influence in the early stages of VRP, followed by ”traveling salesman problem, tabu search algorithm, etc.”. New burst words subsequently appeared in different periods, along with new research hotspots, highlighting the diversification and development of the research content. The novel keywords emerging from 1990 to 2009 were mainly related to path optimization algorithms, including ”heuristic algorithm, time window, combinatorial optimization, tabu search algorithm, integer programming, and column generation”; from 2010 onward, the burst words were mainly”inequality, constraint, CO₂ emission, last-mile delivery, UAV, urban area, optimization model, etc.”. The keyword burst knowledge map basically reflects the current research hotspots and development trends of VRP. According to the citation mutations from 1994 to 2020, it can be found that ”last-mile delivery”, ”UAV”, ”urban area”, and ”optimization model” are the current and future research hotspots of VRP.

4. Discussion

4.1. VRP Model and Its Variations

4.1.1. Taxonomy of VRP Model

After Dantzig and Ramser first proposed the truck scheduling problem in 1959, Clarke summarized it as the vehicle route optimization problem commonly encountered in logistics and transportation fields by adding some constraints in line with reality [1,48]. With the importance of VRP in logistics and transportation fields, since the 1990s, research on the VRP problem has attracted increasing attention and achieved rich results. However, the changing demands and the complexity of the real world have led to many variants of VRP models. Thus, some studies classified VRP models and determined the development trends by combing the literature.

By distinguishing and grouping studies according to their attributes or characteristics of knowledge, publications can be effectively integrated to further reflect the field, as well as summarize existing theories, models, and findings [49]. Bodin first classified static routing and scheduling problems [50]. Desrochers et al. analyzed the existing routing and scheduling literature, proposed a three-layer classification scheme, and developed a vehicle routing and scheduling problem model and algorithm management system [51,52]. Psaraftis clearly defined the dynamic vehicle routing problem according to the available dynamic transportation models, and proposed some future research directions [53]. Subsequently, Eksioglu et al. proposed a comprehensive classification framework for VRP models and briefly described VRP and its variants [54]. Braekers et al. classified the VRP articles published between 2009 and 2015, and then revised and extended the classification framework proposed by [31]. Similarly, Tan et al. provided a structured classification of VRP as a function of solution and problem attributes from the literature published between 2019 and August 2021, summarizing the recent trends of VRP model variants and their solution methods [19].

Figure 13 provides a three-level classification of VRP models following the research of Eksioglu, Braekers, and Tan [19,31,54]. The first-level classification has four main features (scenario features, problem physical features, information features, and data features), each of which has its own detailed categories and subcategories. According to [31], we merged 2.5.1 and 2.5.2 in [54] into the subcategory “depending on”. For the problem physical feature category, “3.3 customer geographical location” was deleted, and 3.6.1 was adjusted. In addition, due to the different requirements of VRP, it has different objectives, such as focusing on travel time, distance, vehicle number, cost related to being late, cost related to risk or danger, any other type of objective, or a combination of these. In order to provide an objective discussion, we adjusted “3.11 transportation cost” to “objective”, then and adjusted the subcategories to “single objective” and “combination objective” according to the diversity of objectives. Thus, the classification in Figure 6 sufficiently covers the diversity of existing VRP models. As the retrieval data cover a long period, the classification was not verified in detail. Specific verification methods can be found in [19,31,54].

4.1.2. VRP Variants

Figure 13 describes four characteristics of VRP models and their detailed classification, namely, scenario features, physical features, information features, and data features, which basically cover existing VRP model variants. Most of these features are generally considered separate in reality, or together with a limited number of other features, whereas complex combinations have been scarcely researched. In order to make the VRP model more reflective of reality, the effective combination of these features can be considered, and methods for solving these problems can be developed in the future. Below, a brief overview of some variants that often appear together is provided, along with some features.

The capacitated vehicle routing problem (CVRP) is a classic variant of the vehicle routing problem [55,56,57]. The CVRP model has a distribution center, a number of geographically randomly distributed delivery points, and vehicles from the distribution center to provide services for these delivery points. To find the best route for vehicles to travel from a large number of distribution plans while reducing the delivery time and cost, three conditions must be met: ① the maximum load of each vehicle must be greater than or equal to the maximum demand on each delivery route; ② the maximum distance that each vehicle can travel must not be less than the maximum distance required for each delivery route; ③ there is only one delivery vehicle at the distribution center. As a classic VRP variant, capacity constraints are generally implicit in other VRP variants [21].

In the vehicle routing problem with backhauls (VRPB), customers can request or return some goods. Therefore, each route is a mixture of line transportation and return transportation customers, and the return customers are usually visited after the line transportation. By allowing the empty truck to return to pick up the goods after delivery, cost saving can be achieved [58,59,60]. The combination of linehauls and backhauls has proven very valuable to the logistics and distribution industry (e.g., the concept of “milk run”) through the successful application of VRPB [61].

The vehicle routing problem with time window (VRPTW) is an extension of the basic vehicle routing problem, with the additional constraint of customers having time windows for the delivery of goods. It can be seen as a combination of the vehicle routing and scheduling problem, which often appears in many real-world applications. In this type of problem, given the earliest and the latest time for vehicles to arrive at a destination, vehicles must arrive within the specified time window so as to not incur additional penalty costs. It was originally proposed by Solomon and Desorios in 1987 [62]. Time windows are divided into three categories: soft time windows (customers accept services that exceed the upper limit of the time window, but impose certain penalties on the delivery party), hard time windows (customers have strict requirements for time windows and do not accept services that exceed time windows), and mixed time windows [63,64,65]. The hard time window algorithm is simple to implement, only needing basic time constraints without considering flexibility; this approach can reliably avoid premature arrivals and late departures, thereby reducing the waiting time of customers. Soft time windows consider the flexibility of time windows to better satisfy the service quality requirements of customers. However, in real life, the setting of soft and hard time windows is relatively one-sided, the flexibility is poor, and it is impossible to fully express the service requirements of customers at different times, contributing to customer satisfaction. A fuzzy soft time window is more appropriate as penalties are difficult to determine and the financial losses caused by the violation of time windows can be long-term.

The dynamic vehicle routing problem (DVRP) refers to the planning and design of the driving route of mobile logistics vehicles under certain dynamic constraints, in order to optimize it (e.g., shortest distance, lowest cost, fastest speed, and least number of vehicles used). The dynamic constraints here refer to various parameters that affect the dynamic driving route, including the supply/demand of goods, the location of loading/unloading points, the customer’s restriction with regard to arrival time of the vehicle, the real-time traffic conditions, the current location, and the cargo of the logistics vehicle [66]. The research on DVRP can be traced back to the 1980s, where it was first proposed by Psaraftis (1988) [66]; further studies on the characteristics of DVRP were then conducted by Bertsimas [67,68] and Psaraftis et al. [53]. In the literature, there are two main classes of DVRPs: deterministic and stochastic. In deterministic versions, some or all requests are unknown and only received during the execution of the vehicle route, while stochastic DVRPs randomly receive information during the execution of the vehicle route [3].The main features of the current DVRP are as follows: the system can receive all kinds of dynamic information in real time, including the needs of new customers, the adjusted demands and service times of old customers, and occasional traffic congestion on the road; the time when the system receives dynamic information such as customer demand and real-time traffic conditions is random and unpredictable, and the content of dynamic information, such as the number of new customers, service time window, and traffic congestion degree, is uncertain; after receiving various dynamic information, the system needs to replan and the vehicle route in real time according to the determined route update strategy; after receiving all dynamic information, the system needs to process it so as to replan and update the vehicle route through rapid calculation. In DVRP, conditions such as traffic congestion or vehicle failure result in the vehicle traveling more slowly than usual, whereby updates to the vehicle route can lead to increased vehicle fuel consumption and carbon emissions [3]. Existing research on the DVRP model mainly considers dynamic information such as dynamic demand parameters and dynamic travel time, as well as constraints such as capacity or time window. However, with the rapid development of smart logistics, new challenges have emerged, such as DVRP model research based on multiple dynamic parameters, dynamic targets, and multiple constraints, as well as research on green DVRP models considering energy conservation and emission reduction [4].

The VRP with pickups and deliveries (VRPPD) considers the possibility of customers re-turning certain goods. This constraint complicates the planning problem and may lead to low utilization of vehicle capacity, increased travel distance, or the use of more vehicles. Therefore, it is usually necessary to consider a limited case, e.g., all delivery demands start from the depot, all pickup demands must be returned to the depot, and there is no exchange of goods between customers. One option is to relax the restriction that all customers must be visited exactly once. Another common simplification is to consider that each vehicle must deliver all goods before picking up any goods. The purpose is to minimize the number of vehicles and travel time, as well as limit the capacity of vehicles to transport goods to and from the depot [69,70,71].

In the vehicle routing problem with simultaneous delivery/pickup (VRPSDP), the same customer may request both delivery from the depot and pickup of other goods for delivery to the depot [72,73]. Yu et al. proposed a vehicle routing problem for simultaneous pickup and delivery and occasional drivers (VRPSPDOD) based on the importance of product returns and the emerging concept of obtaining some compensation in pickup and delivery activities, with the goal of minimizing the total travel cost of operating regular vehicles and the total compensation paid to temporary hired drivers [74]. Hosseini-Motlag et al. proposed a multipath traffic-covering pollution routing problem (PRP) with simultaneous pickup and delivery. In the case of traffic congestion, they determined the path with the lowest cost, extending the traditional PRP model [75].

The multi-depot vehicle routing problem (MDVRP) is an extension of the basic vehicle routing problem, which refers to the service of multiple users by multiple depots at the same time, requiring an appropriate arrangement of the vehicles and routes of each depot to ensure that the total transport cost is minimized while satisfying the needs of all users. MDVRP is one of the most challenging variants of VRP; in combination with other features, it greatly enriches the study of VRP models [76,77,78,79].

The multi-echelon vehicle routing problem (MEVRP) refers to the establishment of one or more logistics centers between enterprises and customers, with the goal of optimizing vehicle routes to reduce the total transportation cost and the number of vehicles used in the entire distribution system [80]. Tadaros et al. introduced a hierarchical multi-switch multi-echelon vehicle routing problem, a new variant of the well-known vehicle routing problem based on a real-world problem, including a single warehouse, intermediate facilities, and two different fleets consisting of homogeneous original and homogeneous local vehicles [81]. MEVRP is widely used in newspaper and magazine distribution, e-commerce goods delivery, etc.

The time-dependent vehicle routing problem (TDVRP) considers how to arrange vehicle delivery routes in a network environment where the travel time of a section changes with the departure time in order to achieve certain optimization goals. The travel time between two customers or between a customer and a distribution center depends on the distance between each point and the time of day. The aim is to assign vehicles to customers and arrange vehicle routes so that the total route time is minimized [82,83]. The time dependence can be combined with other characteristics (such as the time window and pollution path) to construct VRP variants that meet different practical scenarios [17,40]. The time-dependent VRP is applicable to green transportation research aimed at achieving low cost, low fuel consumption, high efficiency, and high utilization of roads. For example, Franceschetti et al. constructed a time-dependent VRP using a comprehensive emission function that included factors such as load and vehicle speed in the pollution routing problem, effectively dealing with carbon emissions in transportation [84].

The split delivery vehicle routing problem (SDVRP) was first introduced by DROR et al. [85], differing from the traditional VRP in that it removed the constraint that each demand point could only be visited by one vehicle, by splitting the order into several parts; accordingly, one customer can be visited by multiple vehicles. Interested readers can refer to [86,87,88,89].

In recent years, people have shown increasing interest in the environmental VRP. As an extension of the VRP, the green vehicle routing problem (GVRP) is gaining more and more attention. Bektas and Laporte proposed the pollution routing problem (PRP) in 2011 [90], while Erdogan and Miller-Hooks were the first to propose the GVRP in 2012 [5]. Currently, the GVRP mainly includes three categories: fuel vehicle routing problems that take into account economic and environmental benefits, vehicle routing problems in reverse logistics, and the problem of vehicle routing for new energy vehicles (electricity, hydrogen, natural gas, etc.) with cost minimization as the goal of optimization [6,91]. They are all optimized to reduce operating costs, reduce energy consumption and carbon emissions, and provide customer satisfaction under the premise of meeting service requirements and other constraints. In the green vehicle issue, economic factors (operating costs such as route operating costs and fuel consumption costs) and environmental goals (carbon emissions) often conflict with each other; thus, in order to achieve sustainable development, multiple methods must be used to balance these conflicting goals when building the GVRP model. For example, a multi-objective optimization model can be built to simultaneously consider vehicle operating costs, increased vehicle utilization, and the impact of carbon tax and carbon trading policies on the objective function to reduce carbon emissions [92,93,94,95]. Constraints can be used to limit conflicts between economic and environmental goals; for example, upper limits on environmental indicators such as carbon emissions and energy consumption can be set to ensure that companies pursue economic goals without having an excessive impact on the environment. In addition, some technical means are used to reduce the conflict between economic factors and environmental goals; for example, intelligent scheduling systems can be used to optimize vehicle driving routes to reduce energy consumption and carbon emissions, while implementing energy-saving and emission-reducing equipment, such as new energy vehicles and solar panels, to reduce vehicle operating costs and environmental pollution [96]. In the context of the GVRP, the constraints of actual fuel consumption and emissions, as well as the new energy VRP, have been widely studied in the recent literature, leading to the development of new variants [90,97,98]. Recently, Sabet et al. provided a detailed summary of the research status, variants, and solution algorithms for GVRP, as well as proposed future research directions and challenges for GVRP [91]. In previous studies, we found that uncertainty parameters (such as uncertain travel time) were often ignored when constructing GVRP models. Most TDGVRP models are based on static formulations, whereas dynamic models are less studied. Most GVRP studies were based on a single objective function, whereas multi-objective optimization is less applied. The GVRP and alternative energy VRP based on new situations such as crowdsourcing services deserve more attention; readers can refer to the specific GVRP variants and their solution algorithm classification in [6,7,40,91].

The open vehicle routing problem (OVRP) was first proposed by Schrage in 1981 [99]. In the OVRP, vehicles do not need to return to the central depot after visiting the last customer. If they are required to return to the central depot, they must return along the original route. The OVRP usually has two optimization objectives: minimizing the number of vehicles used and (given the number of vehicles) minimizing the total distance (or sometimes time) traveled. It has important application value in economic activities with outsourcing characteristics in reality, such as newspaper delivery and milk delivery [100,101,102].

The multi-type vehicle routing problem (MTVRP) is an extension of the vehicle routing problem. According to whether the model of the vehicle is the same or not, the vehicle routing problem can be divided into the homogeneous VRP or heterogeneous vehicle routing problem (HVRP). The single-model problem assumes that the models of the vehicles are the same, i.e., they have the same maximum load carrying capacity and maximum driving distance, as well as the same fixed costs and variable costs [103,104], whereas the multi-type problem is different. For example, the heterogenous fleet VRP (HFVRP) is closer to the actual situation, in which a fleet composed of different types of vehicles provides delivery services to customer points, usually with different maximum load carrying capacity or different usage costs [105,106,107].

Two-dimensional and three-dimensional VRP (xDVRP) and two-dimensional loading VRP (2L-CVRP) introduce capacity constraints to the CVRP, combined with a two-dimensional packing problem, to meet the needs of all customers [108,109,110]; three-dimensional loading VRP (3L-VRP) is a combination of the three-dimensional loading problem and vehicle routing problem, where the same conditions as 2L-VRP are generally applicable [111,112,113].

In the collaborative VRP (ColVRP), multiple distribution centers of service providers cooperate to improve the efficiency of their logistics operations, ensuring that customers receive services on time while reducing operating costs [77,114,115]. Collaborative VRP research is likely to increase in the future, as cooperation among logistics companies yields huge benefits [116]. Chinh et al. proposed a new collaboration strategy that requires less coordination effort among logistics service providers, thus helping to reduce the total costs and increase overall truckload in urban last-mile logistics [117]. MacLachlan et al. provided a novel solution construction procedure that generates solutions to the uncertain capacitated arc routing problem within a collaborative, multi-vehicle framework, and then used a genetic programming hyper-heuristic algorithm to evolve the routing policy, clearly showing the advantage of vehicle collaboration in handling an uncertain environment [118]. Mancini et al. introduced the collaborative consistent VRP with workload balance and analyzed the potential benefits of multi-period collaborations [119]. Through research on the collaborative capacitive electric vehicle routing problem, it has been shown that the collaborative strategy can significantly reduce the total cost and total energy consumption, while significantly improving the customer service level and vehicle utilization [120]. Wang et al. proposed the collaborative multicenter vehicle routing problem with time windows and mixed deliveries and pickups, where open–closed mixed-vehicle routes were constructed through collaboration and transportation resource sharing [121].

The consistent VRP (ConVRP) is a relatively new problem variant, which has received relatively less attention. This is mainly because, for most logistics companies, there is no stable delivery in a period of time, and it is not possible to plan routes for multiple days. Groer et al. (2009) first proposed the concept of ConVRP, which requires the same driver to visit the same customers at roughly the same time every day [122]. From the driver’s perspective, consistency includes customer consistency, arrival time consistency, and route consistency [122,123,124].

The periodic vehicle routing problem (PVRP) is an extension of the VRP in terms of time, which involves constructing vehicle routes within a period of time, where the time horizon extends to several periods. Routes are constructed for each period, and each customer is visited once or more times within the time range according to their requirements. The goal of the PVRP is to find a set of routes for each vehicle so that the total traveling cost (such as distance) is minimized, thus satisfying the basic requirements of customer needs and vehicle capacities. Inspired by garbage collection, Beltrami and Bodin (1974) proposed the first periodic routing problem [125], and other scholars further extended the research on this problem [126,127].

In the truck and trailer vehicle routing problem (TTVRP), the fleet consists of a known number of single-box trucks and trailers, which can be separated from each other. Through the cooperation and coordination of trucks and trailers, Chao and Scheuerer established the trailer transport path planning model and used the improved tabu search algorithm to solve the model [128,129]. Subsequently, a series of TTVRP models and algorithms were studied [130,131].

4.2. Solutions for VRP

From the perspective of graph theory, the TSP and VRP problems essentially necessitate finding a Hamiltonian cycle with the minimum weight in a completely weighted undirected graph. Since the feasible solution of the problem is the permutation of all vertices, a combination explosion occurs, which increases the number of vertices, making it an NP-complete problem. Due to the complex constraints involved in real-world problems, advanced algorithms are needed to solve VRPs in complex and ever-changing environments. A large number of different VRP solving strategies have been proposed in the literature, including exact, classic heuristic, metaheuristic, hyper-heuristic, machine learning, and hybrid methods. Exact methods can obtain the global optimal solution; however, due to the NP-hard complexity of VRP and its variants, solving a large-scale problem optimally is very time-consuming. Thus, exact methods are usually only applicable to small-scale problems. Heuristic and metaheuristic algorithms have no restrictions in terms of the size of the problem; they can effectively handle a large number of constraints and obtain near-optimal solutions within an acceptable time frame. Heuristic and metaheuristic algorithms are commonly used today to solve the VRP and its variants, but there is still a great challenge in how to select a specific algorithm due to the high sensitivity of multiple parameters and their configurations [132,133,134]. Hyper-heuristic algorithms are recently proposed concept models for solving complex optimization problems with advantages such as generality and efficiency. As effective advanced autonomous search methods, hyper-heuristic algorithms can automatically select, combine, or generate multiple simple low-level heuristic algorithms to solve complex optimization problems. In addition, with the development of machine learning and deep learning, algorithm design by machines can substantially save time in solving combinatorial optimization problems, while also producing better solutions than traditional methods [135,136,137].

According to the WoS retrieval results and the results of the literature metric analysis, the current VRP problem-solving algorithms were classified in terms of the methods used in [19,116,138] (Figure 14). Since classical heuristic methods do not allow intermediate solutions to deteriorate in the search for optimal solutions, they often become stuck in local optima [48,139,140], whereas metaheuristic methods incorporate mechanisms to avoid this issue [141,142]. Therefore, in [54], Braekers et al. suggested dividing heuristic methods into classical methods and metaheuristic methods [31]. In addition, considering the recently emerging combined optimization algorithms in VRP research, we added hyper-heuristic algorithms and machine learning algorithms. Compared with the above-described algorithms, hyper-heuristic algorithms and machine learning algorithms are relatively new in the VRP research. Thus, only one level of classification is proposed, with further elaboration expected in the future.

4.2.1. Exact Algorithms

Exact algorithms can obtain the optimal solution of the VRP; they mainly include branch and X (X: cutting, constraint, pricing, etc.) methods [144], dynamic programming [145], constraint programming [138], mixed-integer linear programming, column generation methods [146], and network flow algorithms [147] (Figure 14). Specifically, each algorithm has its own applicable scope and characteristics. The branch and bound method can implicitly explore the solution space [148]; it can be combined with the cutting plane to form the branch and cut algorithm [149], or combined with column generation to form the branch and price algorithm [150]. Dynamic programming decomposes complex problems into simpler subproblems to obtain optimal solutions. Constraint programming is a model that uses constraint conditions to associate different variables. The column generation algorithm simplifies the VRP, introduces a dual-variable vector to relax the problem, and determines the optimal solution by calculating the minimum marginal cost of columns. Branch and X methods are commonly used for obtaining exact solutions to the VRP [151,152].

Exact algorithms have many advantages in solving VRP problems. First, they can verify whether the mathematical model of the VRP problem is correct, which can ensure the correctness and reliability of the VRP problem. Second, for small-scale or medium-sized VRP problems, they can find the global optimal solution, which is reliable. Third, they can provide an exact benchmark to test the accuracy and efficiency of the heuristic algorithm. In addition, some VRP variants need to be modeled by nonlinear programming models, which increases complexity; therefore, the nonlinear optimization model needs to be linearized to make it more suitable for solving using exact solution algorithms [153].

In general, exact algorithms are based on strict mathematics models, and their solutions are usually superior to those of other algorithms. However, their introduction results in the solution process becoming more complicated, whereby the computational complexity and running time may become impractical upon increasing the scale of the problem. Therefore, the optimal solution can only be obtained when the number of customers is small, and the transportation network is simple. In order to improve the efficiency and quality of the solution, as well as obtain an approximate or satisfactory solution of the problem at a reasonable computational cost, exact algorithms are generally combined with other methods when solving the problem [154,155,156,157].

4.2.2. Classical Heuristic Algorithms

Compared with the solution process and time consumption of exact algorithms, classical heuristic algorithms give an acceptable and feasible solution to the combinatorial optimization problem within an acceptable range, albeit not the optimal solution, thereby solving large-scale VRP problems [70,158,159]. Heuristic algorithms solve problems through the introduction of past experience and experimental analysis, i.e., by means of intuitive induction or exploration methods. Heuristic algorithms require analysts to use their own perception and insight to seek connections among relatively specific models and algorithms related to the research problem, thus obtaining inspiration to discover suitable approaches to solve the problem. In other words, heuristic algorithms improve the known feasible solutions according to some heuristic information and obtain a relatively satisfactory solution through several iterations. Compared with exact algorithms, heuristic algorithms cannot guarantee the global optimal solution. Most of these algorithms can be easily extended to deal with various constraints encountered in practical applications, generally obtaining satisfactory solutions in a short time, while being relatively simple to implement. For large-scale VRPs, heuristic algorithms have higher robustness and feasibility. As shown in Figure 14, heuristic algorithms can be mainly divided into three categories: construction algorithms, two-phase algorithms, and improvement algorithms. Construction algorithms can be divided into four categories (the C–W savings algorithm, nearest neighbor method, recently inserted method, and scanning algorithm) [48,62,160]. Two-phase algorithms include the Cluster first/route second [161] and route first/cluster second methods [162]. Improvement algorithms include inter-route and intra-route methods [40,116]. For detailed information on specific algorithms, the readers can refer to [134,163,164].

4.2.3. Metaheuristic Algorithms

Metaheuristic algorithms improve heuristic algorithms by combining random algorithms and local search algorithms. Constructed on the basis of intuition or experience, metaheuristic algorithms can provide a feasible solution to a problem within an acceptable cost (in terms of computing time and space), and the deviation between the feasible solution and the optimal solution may not be predictable in advance [137,141]. Compared with heuristic algorithms, metaheuristic algorithms improve the quality of the solution through a more comprehensive and thorough search process, with scholars making substantial progress in this field. According to the number of parallel objective solutions in each search iteration process, metaheuristic algorithms can be divided into two categories: single-solution-based and population-based [143]. Single-based heuristics can be further divided into seven categories (SA, LNS, TS, ILS, GLS, VNS, and GRASP), with SA being further divided into SA-NM, SA-TA, and SA-MA, and LNS being further divided into ALNS and VLSN. Population-based algorithms can be divided into two categories (EC and SI), with EC being further divided into 12 subcategories, and SI being further divided into 10 subcategories. According to the survey results of [143], the literature on the TS and VNS of single-solution-based metaheuristic algorithms was predominantly published in 2009–2015, whereas the LNS, SA, ILS, and GRASP were relatively less applied, and GLS was rarely used. In terms of population-based metaheuristic algorithms, GA was the most used EC algorithm, whereas MA was the least used; for SI algorithms, ACO and PSO were the most used, whereas BFO and AIS were almost never used. For detailed descriptions of each algorithm, the readers can refer to [134,165,166].

In single-solution heuristic algorithms, such as simulated annealing algorithms, tabu search algorithms, and neighborhood search algorithms, each round of search only starts from one solution. Tabu search (TS) is a global neighborhood search algorithm that simulates the characteristics of humans with memory. The tabu search algorithm was first proposed by Glover in 1986, and many scholars have since perfected this algorithm, becoming one of the most widely used heuristic algorithms in the solution of combinatorial optimization problems [141,167,168,169,170]. It avoids detours through local neighborhood search mechanisms and corresponding tabu rules, while releasing some prohibited excellent states according to the breaking level so as to ensure diversified and effective exploration, and then achieve global optimization. Its search path mainly consists of initial solution generation, neighborhood structure construction, and tabu table design. Good initial solutions can improve the search space, while neighborhood structures can enhance the search ability. The tabu table can delay the algorithm from falling into local optima and enhance the optimization ability. Variable neighborhood search (VNS) is an improved local search algorithm that alternately searches using different action-based neighborhood structures to achieve a good balance between concentration and dispersion. The main idea of the variable neighborhood search algorithm is to use multiple different neighborhoods for systematic search. The smallest neighborhood search is initially used, and, when no further improvement to the solution is possible, a slightly larger neighborhood is applied. Upon any improvement to the solution, the smallest neighborhood is reintroduced; otherwise, an even larger neighborhood is applied [171,172,173]. Simulated annealing, greedy randomized adaptive search, and iterative local search algorithms all use strategies to avoid local optima [143].

Population-based metaheuristic algorithms start each round of search from a relatively independent and parallel set of solutions, both of which need to be retained according to rules meeting the requirements of the optimal solution. The genetic algorithm is the most common metaheuristic algorithm based on population evolution, which exchanges information by means of reproduction, mutation, competition, etc. among individuals in the population, thus gradually approaching the optimal solution of the problem. Genetic operations on individuals are mainly realized by three basic genetic operations: selection (reproduction), crossover, and mutation. The advantages of the genetic algorithm are that the solution is stable, and the computational efficiency is high; however, it suffers from weak local search ability, and it takes a long time to reach the optimal solution upon approaching it. In addition, if the fitness function is not properly selected, the genetic algorithm often converges to local optima, failing to identify the global optimum [174,175,176]. Genetic algorithms can be improved in four aspects: individuals and populations, selection operations, crossover operations, and mutation operations, thereby optimizing the population and addressing the drawbacks of early maturity and local optima; an example of an improved GA is MA [177,178,179]. The ant colony algorithm and particle swarm algorithm are the most commonly used swarm intelligence algorithms. The genetic algorithm originated from a computer simulation of biological systems [174]. It is an efficient, parallel, and global search method that simulates the evolution mechanism of the natural biological world. It draws on Darwin’s evolution theory and Mendel’s genetic theory. In essence, it is a global search and optimization method that can automatically acquire and accumulate knowledge about the search space while adaptively controlling the search process to obtain the best solution.

The ant colony algorithm is a swarm intelligence biomimetic algorithm, inspired by the behavior of ants foraging in nature; it involves the mutual cooperation of a group of unintelligent or slightly intelligent individuals (agents) [180]. The ant colony optimization (ACO) imitates the ant path-finding mechanism. Unlike other metaheuristic algorithms, it applies distributed construction of solutions. When an ant chooses the next step, it needs to refer to the accumulated experience (pheromone distribution) of the whole population, as well as the heuristic information. When an ant successfully builds a complete solution, it enhances the accumulated experience of the group according to the quality evaluation of the solution. The biggest advantage of the ant colony algorithm is its strong search ability for optimal solutions. In addition, it has the advantages of a simple principle, parallelism, robustness, and easy application. Its disadvantages are the long search time and low computing efficiency. In addition, the setting of parameters (such as the pheromone volatilization factor) has a great influence on the results, whereby an improper setting leads to a local optimum and stagnant phenomenon, preventing a further search for the optimal solution. In order to address this drawback, the ACO can be improved in terms of individuals and populations, feasible solution construction, and pheromone update.

Particle swarm optimization (PSO) is a global stochastic optimization algorithm based on collective intelligence, inspired by the foraging behavior of birds. By simulating this behavior, the motion of the whole group in the problem-solving space evolves from disorder to order through the sharing of information by individuals in the group, so that the group can achieve the best goal. Instead of the crossover and mutation operations of the genetic algorithm, PSO seeks the global optimal solution by following the current known optimal value [181,182,183]. Compared with other optimization algorithms, PSO has few parameters to adjust, has a fast convergence speed, and is simple and easy to implement. However, due to the lack of a mutation mechanism like the genetic algorithm, PSO falls easily into local optima, and its convergence speed is slow in later iterations. Some other parameters can be introduced or PSO can be combined with other intelligent optimization algorithms, such as the genetic algorithm (GA), immune algorithms, and simulated annealing algorithms, to help particles jump out of local optima and improve convergence speed.

4.2.4. Hyper-Heuristic Algorithms

Despite the success of heuristic algorithms and other search techniques in solving re-al-world computational search problems, there are still difficulties in applying them to new instances of similar problems, mainly in terms of parameter adjustment and algorithm selection. Hyper-heuristic algorithms generate high-quality solutions by manipulating a series of low-level heuristics (LLH) through high-level heuristics (HLH) [135,184]. These new heuristics are used to solve various NP-hard problems [185]. Hyper-heuristics consist of two levels. At the problem domain level, domain experts need to design a series of low-level heuristics according to the specific characteristics of the problem, while providing the problem definition, problem representation, initial solution, evaluation function, etc. In the high-level strategy, intelligent computing experts design efficient control and management mechanisms, at each decision point, using the algorithm set and problem feature information (usually the return value of the evaluation function) provided by the problem domain to construct new or select existing heuristics, and then applying them to specific problem instances. Due to the strict domain shielding between the two levels, as long as the algorithm sets and defines the problem, along with modification of the problem representation, initial solution, evaluation function, etc., the hyper-heuristic algorithm can be easily transplanted to a new problem.

Existing hyper-heuristic algorithms can be roughly divided into four categories according to the mechanism of high-level strategy: random-based, greedy-based, metaheuristic-based, and learning-based [135]. Random-based hyper-heuristic algorithms randomly select LLH from a given set to form new heuristic algorithms, characterized by a simple structure and easy implementation. At the same time, these hyper-heuristic algorithms are often used as benchmarks to evaluate the performance of other types of hyper-heuristic algorithms. These hyper-heuristic algorithms can be further divided into pure random and random with late acceptance. Greedy-based hyper-heuristic algorithms select LLH that maximize the improvement of the current (problem instance) solution when constructing new heuristic algorithms. Since all LLH need to be evaluated in each instance when selecting LLH, the execution efficiency of this method is lower compared to random-based hyper-heuristic algorithms. Metaheuristic-based hyper-heuristic algorithms use existing metaheuristic algorithms (as a high-level strategy) to select LLH. Metaheuristic algorithms can be further divided into tabu search-based, genetic algorithm-based, genetic programming-based, ant colony algorithm-based, and GRASP with path relinking-based [136,186,187]. Learning-based hyper-heuristic algorithms use certain learning mechanisms to decide which LLH to adopt according to the historical information of various LLH when constructing new heuristic algorithms. According to the source of LLH historical information, these hyper-heuristic algorithms can be further divided into three types: no learning, offline learning, and online learning. The no learning mechanism randomly determines the combination and execution sequence of low-level heuristic algorithms during calculation; offline learning usually divides the instance set into training instances and unsolved instances, whereby training instances are mainly used to accumulate historical information of LLH, while unsolved instances can determine which LLH to adopt according to these historical information; online learning refers to the historical information of LLH accumulated during the solution of the current instance [135,184].

Cowling first described the hyper-heuristic algorithm as “a heuristic for finding heuristics” [188,189]. Burke provided a more precise definition of hyper-heuristics as providing “a higher-level heuristic approach to various combinatorial optimization problems by managing or manipulating a series of low-level heuristics” [190]. For the CVRP, Garrido et al. proposed a hill-climbing-based hyper-heuristic and an evolutionary based hyper-heuristic, where the former constructs and improves the partial solutions by continuously applying constructive–perturbative pairs of low-level heuristics [187], while the latter only contains one heuristic component, using a recombination operator and eight mutation operators to generate new individuals, with the individual population evolving according to a steady-state evolutionary model [191]. Mlejnek et al. proposed a new evolutionary construction hyper-heuristic method called HyperPOEM, which allows autonomous search of various low-level heuristic structured spaces in order to find suitable combinations and provide good solutions. The HyperPOEMS technique outperforms the two hyper-heuristic approaches proposed by Garrido et al. [186]. Sim et al. proposed a novel genetic programming metaheuristic (GP MHH) algorithm to solve the vehicle routing problem, consisting of two phases; the first phase uses a novel genetic programming to evolve a constructive heuristic population, while the second phase perturbs the solutions generated in the first phase using a disturbance hyper-heuristic algorithm [192]. Leng et al. proposed a novel multi-objective hyper-heuristic (MOHH) algorithm to obtain Pareto solutions for the dual-objective cold-chain routing problem considering the environmental impact; three selection strategies were introduced to enhance the performance of the MOHH, and the proposed algorithm was verified to outperform several existing multi-objective evolutionary algorithms [193]. Ortiz-Aguilar et al. selected low-level heuristics in the offline learned hyper-heuristic method by way of meta-learning to improve the performance of offline hyper-heuristic algorithms; through verification using the CVRP test, the results show that the proposed method can improve the performance of offline hyper-heuristic algorithms [185]. Kalatzantonakis et al. proposed a bandit VNS hyper-heuristic algorithm, which was tested on CVRP benchmark instances, and then compared it to the traditional general variable neighborhood search (VNS) metaheuristic, revealing that the computational speed and the quality of solutions were significantly improved [194]. Rodríguez-Esparza et al. proposed a hyper-heuristic (HH) approach, named reinforcement learning hyper-heuristic adaptive simulated annealing (HHASARL), constituting a multi-arm bandit approach and an adaptive simulated annealing (SA) metaheuristic to solve the electric vehicle routing problem (EVRP). The implemented HH scheme improved several of the known minimal solutions and achieved the best averages in some high-dimensional instances [14].

4.2.5. Machine Learning

Metaheuristic algorithms generate extensive data, such as the fitness value of the solution, evolutionary trajectory, and local optimal solution, in the process of obtaining an approximate solution. These data may carry some useful knowledge such as the attributes of the solution, as well as the performance and priority of the operator. However, metaheuristic algorithms do not take advantage of this hidden knowledge. Machine learning can extract useful knowledge from the generated data in the search process to make metaheuristic algorithms more intelligent, thus improving the quality and robustness of the solutions [15].

Machine learning algorithms are methods that automatically analyze data to obtain rules, and then use these rules to predict unknown data. Algorithms and statistical methods are applied to give computers the ability to “learn” from data and improve their performance in solving problems without explicitly programming for each problem. ML algorithms can be divided into three categories: supervised learning, unsupervised learning, and reinforcement learning [195]. Reinforcement learning (RL) does not require a supervised signal to learn, but relies on feedback signals from the individual (agent) in the environment (environment) to correct the individual’s state and action according to the feedback signals, so that the individual can gradually realize the maximization of rewards, thus achieving strong self-learning ability. RL algorithms can be divided into two categories: model-based learning and model-free learning [196]. Model-based learning has prior knowledge of the environment, which can be optimized in advance; model-free learning is inferior to the former in terms of training speed, but is easier to implement and can quickly adjust to a better state in real scenarios. Common model-based learning methods for solving the VRP include dynamic programming; model-free learning algorithms mainly include value-based temporal difference algorithms [197], Q-learning algorithms [198], DQN algorithms [199], policy-based REINFORC algorithms [200], value and policy combined actor–critic algorithms, and advantage actor–critic algorithms [201].

Deep reinforcement learning (DRL) combines the feature extraction capabilities of deep learning and the decision-making capabilities of reinforcement learning (RL), which directly allows making the best decision output on the basis of multidimensional input data. It is an end-to-end decision control system, which is widely used in dynamic decision making, real-time prediction, simulation, game playing, etc. It interacts with the environment in real time, takes the environment information as input to obtain experiences of failure or success, and updates the parameters of the decision network, thus learning the optimal decision [11,196,202]. In recent years, a large number of studies have been conducted to develop new learning algorithms using DRL methods to automatically solve path problems in order to realize dynamic and efficient solutions. Yang et al. classified DRL methods for solving the VRP into four categories: PN, GNN, transformer, and hybrid models [203]. Czuba et al. discussed the application of machine learning in VRP problems from several aspects, including pointer networks, graph neural networks, recursive neural networks (RNNs), deep reinforcement learning (DRL) methods, and heuristic search algorithms such as Monte Carlo tree search (MCTS) [13]. Most machine learning algorithms have been applied to static and deterministic VRP variants [204,205,206,207], with only a few applications to dynamic VRP problems [11,208]. However, the application of machine learning methods to VRP problems will become more popular [13]. Karimi-Mamaghan et al. comprehensively reviewed machine learning heuristic algorithms for different purposes, such as algorithm selection, fitness evaluation, initialization, evolution, parameter setting, and collaboration, and presented some challenges faced in applying machine learning techniques to heuristic algorithms, such as adjusting additional parameters, data availability, and dynamism [15]. In general, compared with traditional exact algorithms and approximate algorithms, machine learning-related algorithms can be used to solve large-scale dynamic VRPs. They can directly learn features from data, mine the intrinsic relationships among data, and automatically learn the relationship among distribution centers, demand points, and vehicles under different constraints, in order to find the best match or optimal path. After multiple training iterations, they can obtain an efficient and accurate path optimization model, exhibiting great improvement in terms of solution quality, convergence speed and robustness. Further research on machine learning will greatly improve solutions to the VRP problem in the future.

5. Conclusions

Through bibliometric analysis, we provided a comprehensive overview of VRP research papers from 1959 to 2022. Our analysis included bibliometric data on annual publication quantity, subject categories, source journals, countries, organizations, and keywords. The research results showed that the number of VRP research publications has increased significantly since its proposal in 1959. Currently, the number of publications per year exceeds 1000 and is growing, indicating that more and more researchers are contributing to this topic. China, the United States, Canada, Italy, and France are the countries with the greatest research output. The Universities of Montreal, HEC Montreal, and Polytechnique Montreal are institutions with high productivity in VRP research. The European Journal of Operational Research, Computers Operations Research (COR), Transportation Science (TS), and Computers Industrial Engineering (CIE) are the top five journals in terms of VRP publication volume. In terms of publications and influence, Laporte, Gendreau, Wang, Juan, Cordeau, Speranza, Archetti, and Hartl are the most outstanding authors. Through keyword and co-citation network analysis, we tracked the research hotspots and development trends of current VRP research. The visualization map showed that open VRP is currently the largest category in VRP research, followed by stochastic vehicle routing problems. For solving the VRP, metaheuristic algorithms are most widely used, while hyper-heuristic algorithms and machine learning algorithms have received increasing attention. Through a visualization map of the literature, we obtained an understanding of the current status, hotspots, and development trends of VRP research. In addition, on the basis of the latest VRP research status and taxonomy methods, we conducted a comprehensive classification of VRP models and optimization methods, providing detailed knowledge and references for relevant personnel to understand this field.

Bibliometric visualization map analysis provides researchers with a systematic method to understand the evolution of research fields. It helps to identify potential knowledge within the research area and obtain the latest information using various sources of information [23]. However, bibliometric methods have certain limitations. Firstly, for senior researchers, the results of bibliometric methods may be overestimated due to controversial research content, negative citations, etc. [209]. Secondly, a co-occurrence analysis of keywords to track research topics and hotspots may be affected by search data. The setting of search keyword conditions will greatly affect the final search results, and the scope of data coverage will also be limited by the source of search (such as using a single data source). For more information on how to consider the impact of search strategies on search results, readers can refer to Chen (2017). A more in-depth analysis of the research object through appropriate search strategies combined with contextual analysis and other aspects of literature research will provide more insightful analysis results [23]. In this study, we did not set up a complex and detailed search strategy. We only set the “vehicle routing problem” as the search topic without considering related topics such as the “inventory routing problem”, because these related contents were also displayed during visualization map analysis (see Cluster 13# in Table 5). However, they had relatively little impact on our conclusions, since our main focus was on VRP.

Overall, in the past 60 years, VRP research has made significant progress and received increasing attention. With the deepening of research on routing problems and the complexity of real routing problems, more and more variants of VRP have been introduced that are closer to practical situations. The optimization algorithms for VRP models and their variants have also developed rapidly. However, with the rapid development of the economy and the increasing demand for environmental protection, as well as the application of emerging technologies such as machine learning and cloud computing in logistics, the research depth of VRP models and their solving algorithms is still insufficient. There are still many challenges in modeling and optimizing the VRP in terms of theory and methods, and there are many urgent issues that need to be addressed. In terms of model construction, the impact of uncertainty factors in specific VRP variants needs to be taken into account, such as demand, speed, and travel time, which are considered as uncertainty parameters in the GVRP field. However, parameters related to the sustainability of the supply chain network cannot be completely ignored (such as social concerns, customer willingness, and operational risks) [7]. Furthermore, the objective function is a core component of VRP mathematical models and has important implications for future VRP research. Currently, most work focuses on single-objective optimization because it is relatively simple compared to multi-objective functions. There is relatively little research on VRP models with multiple constraints. Therefore, future research can consider multiple objectives in real-life situations and develop efficient algorithms to solve these problems, such as the multi-objective green VRP with mixed constraints. Additionally, unmanned aerial vehicles (UAV) and unmanned ground vehicles or drones are emerging technological solutions for solving last-mile delivery problems and will be increasingly applied in the future (see Cluster 8# in Table 5). How to extend the existing 2D land road network to a 3D land and air traffic network is a future direction for VRP research. Emerging services also have an impact on the development of VRP [91]. For example, crowdsourced transportation services are replacing traditional delivery companies and bringing some challenges to the construction of VRP models (see Cluster 20# in Table 5). Moreover, as the cooperation of logistics service providers will bring huge profits, ColVRP will receive more and more attention, and how logistics service providers cooperate remains a challenge for the future. Lastly, how to consider multiple features in real-life scenarios to construct more realistic models is a challenge that has always received attention in VRP problem research. Most algorithms have high specificity and cannot be applied to other VRP variants; hence, their universality is relatively low. Furthermore, most optimization algorithms attempt to solve large and complex problems; however, the quality of the solution is the most challenging issue. Heuristic algorithms and metaheuristic algorithms are currently the most popular solving methods. Emerging machine learning methods, reinforcement learning, distributed systems, Internet of vehicles (IoV), and new fuel technologies will play an important role in further developing VRP research [91]. How to effectively use machine learning and precise algorithms, hyper-heuristic algorithms, etc., to construct hybrid algorithms to improve the quality and robustness of solutions is a future research topic for VRP problems. Therefore, with the increasing complexity of real scenarios and the rapid development of information technology, how to accurately define VPR model variants and use hybrid algorithms, machine learning, and other related technologies to obtain efficient solution requires further in-depth research on VRP variants and their solution algorithms.